Let $S$ be a surface in 3D described by the equation $x^2 + y^2 + z^2 = 4$. Fill in the rest of the equation of the plane tangent to $S$ at $(1, 1, \sqrt{2})$. $2(x - 1) + $ $ (y - 1) + $ $ (z - \sqrt{2}) = 0$
Answer: The equation for a tangent plane of an implicitly defined surface $F(x, y, z) = 0$ at the point $(a, b, c)$ is: $F_x(x - a) + F_y(y - b) + F_z(z - c) = 0$ [What's the intuition behind the formula?] We can see from the formula that the two values we're missing are $F_y$ and $F_z$. $\begin{aligned} F_y &= 2y = 2 \\ \\ F_z &= 2z = 2\sqrt{2} \end{aligned}$ Here's the completed equation for the tangent plane of $S$ at $(1, 1, \sqrt{2})$ : $2(x - 1) + 2 (y - 1) + 2\sqrt{2} (z - \sqrt{2}) = 0$